Your back pain is probably from lumbar degenerative disc disease (DDD-see website). Martial arts put unusual torsion on the lumbar spine which will be noticed with lower back pain in some individuals who have DDD.
You can help to reduce the torsion with core strengthening exercises. I recommend Pilates as this seems to be the best exercise to work the core muscles. It will take you three months to develop enough strength to be able to withstand these forces in martial arts.
You do not have to worry which side the bulge is the greatest at this time as you don’t currently need interventional care. If time comes that you do need it, the injections should be on the side of greatest pain, not greatest pathology.
Donald Corenman, MD, DC is a highly-regarded spine surgeon, considered an expert in the area of neck and back pain. Trained as both a Medical Doctor and Doctor of Chiropractic, Dr. Corenman earned academic appointments as Clinical Assistant Professor and Assistant Professor of Orthopaedic Surgery at the University of Colorado Health Sciences Center, and his research on spine surgery and rehabilitation has resulted in the publication of multiple peer-reviewed articles and two books.